<h2>Problem 221</h2>
<div style="color:#666;font-size:80%;">13 December 2008</div><br />
<div class="problem_content">
<p>We shall call a positive integer <var>A</var> an "Alexandrian integer", if there exist integers <var>p</var>, <var>q</var>, <var>r</var> such that:</p>

<table class="formula" style="margin-left:50px;">
<tr>
   <td>
      <var>A</var> = <var>p</var> &middot; <var>q</var> &middot; <var>r</var> &nbsp;&nbsp;&nbsp;and&nbsp;&nbsp;
   </td>
   <td>
      <table class="frac">
         <tr><td>1</td></tr>
         <tr><td class="overline"><var>A</var></td></tr>
      </table>
   </td>
   <td>=</td>
   <td>
      <table class="frac">
         <tr><td>1</td></tr>
         <tr><td class="overline"><var>p</var></td></tr>
      </table>
   </td>
   <td>+</td>
   <td>
      <table class="frac">
         <tr><td>1</td></tr>
         <tr><td class="overline"><var>q</var></td></tr>
      </table>
   </td>
   <td>+</td>
   <td>
      <table class="frac">
         <tr><td>1</td></tr>
         <tr><td class="overline"><var>r</var></td></tr>
      </table>
   </td>
</tr>
</table>

<p>For example, 630 is an Alexandrian integer (<var>p</var>&nbsp;=&nbsp;5, <var>q</var>&nbsp;=&nbsp;<img src='images/symbol_minus.gif' width='9' height='3' alt='&minus;' border='0' style='vertical-align:middle;' />7, <var>r</var>&nbsp;=&nbsp;<img src='images/symbol_minus.gif' width='9' height='3' alt='&minus;' border='0' style='vertical-align:middle;' />18).
In fact, 630 is the 6<img src="" style="display:none;" alt="^(" /><sup>th</sup><img src="" style="display:none;" alt=")" /> Alexandrian integer,  the first 6 Alexandrian integers being: 6, 42, 120, 156, 420 and 630.</p>

<p>Find the 150000<img src="" style="display:none;" alt="^(" /><sup>th</sup><img src="" style="display:none;" alt=")" /> Alexandrian integer.</p>
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